Issue 44

V. Reut et alii, Frattura ed Integrità Strutturale, 44 (2018) 82-93; DOI: 10.3221/IGF-ESIS.44.07

where

 2  

 1   

    

'   

 





      c c 1 0 1 0 c c



  

  

    







i

'

,

1, 2

i

i

2

1 

1 

1 

   f 

 

 x d

   ,1 i 



     2







  

1   

, 



, R x d 



, R x d i   ,









K x

0,1, 2

i

i

i

,2







1

1

1

 x R x R x r x i          , ,   , ,     , , , f

are known regular functions,

0,1, 2

i

i

i

,1

,2

 

3     h     

   



 1 1 x 



 1 1 x 

 

 





  







x

x

1 x

1 x

1

1





 



2      h





, x h 

,

1

  2

2

3

3













2      

2

x  

x  

x  

x  









2

2

2

2

2





3

2

4

 

,    . h

h

h

,

1

2

3







2

In this article the misprint in [28] for the coefficients , is corrected. The first singular integral equation in the system (13) is the partial case of the equation with two fixed singularities for the second case. For this equation the transcendental equation was built, that is congruent to the transcendental equation obtained for the quarter plane or, the same, for the problem of an infinity wedge when the angle of openness is pi/2 [32]. The problem for the quarter plane is solved in Appendix A. The roots k  of the corresponding transcendental equation for (13) were found numerically. The generalized method developed in [28] was applied to solve the SSIE (13). According to it the function      is searched in the form 1, 2, 3 i h i 



N

1



  



    

  

  

0 k k  



0

 

 









s

s

,

1;1

(14)

k N k 



k

0

where







Re

   









k

2 

1  





 cos Im ln 1 , sin Im ln 1 , k        

k

, 0 k s

 

0, k N

1







Re

   





k

1  





2 1 k 

k

are the unknown coefficients. It is supposed that the crack is located inside the semi-strip far from the lateral sides. So the unknown functions     1 2 ,       are considered as

2 1 N 

  1;1 ,    i 



    

  

i

2







s

U

(15)

1

,

1, 2

i

k

k



k

0

  k

where  are Chebyshev polynomials of the second kind. The expressions (14)-(15) are substituted in the SSIE (13). The resulting system is solved with the help of the collocation method. The substitution of the founded constants , 0,1, 2, 0, 2 1 i k s i k N    in the formulae (14)-(15) and (11)-(12) U

87